XM are coordinates 4 Scalar functionson spacetimeµ O l 2,3not components of a crater
Uecter function number on function
Um m U Undofy oxm
ComponentsT basisvectors
Codector vectors number or function
W WudxnComponents I basiscooeaters
If we did then Wcv UCO
dxn 04 7 44M ofWW Wendy w v WµVdI WnwDX v V W lov Wu
Tensors
U Un n is a type f tensorcan be acted upon by 1 co vector
W weedXM is a type 9 tensorcan act on 1 vector
type nm teaser can be acteduponby n covectorscan act on in vectors
L teaser T The dead DX nW Tcu v D Trap dx4vddxkv.tw 9ox4
T1pVYufWuwlTIwlTnapdxaxOdxPxo0fyr WntMarsdxa dxBTl V ftp.dxaxodxfxoglyn VT
TrapUP dxd u TL
T H m LThem N
tcml LTIpdxaxodye.IO KMrP9xox09xd
TIaMTfIdfofTIpMdP xuTIoMK µ
not the same
T Ll N ilTrap Nfu components ofacovector
TINI NCT's
9µV metric 9 tenser
g inverse metric f tense
gmgv e.sicomponentsof f tensor
deil
identity tensor
gµTap Trap gudTL Tmpgrug
g
9 d't Sure
n Ed
SiuMM Mmm traceof Mfsindex contraction
T l'drap The
tension1nap MM UM
µ free index takes on 0,1 2 OrB
dip repeated indices summed overfree indices must balance same ineach term
repeated indices eachterm in pairs oneup onedown
Tap MH vieTap MM V
V
TI Mors w
f sane
lower the index
gavTureMH 9mVnewTarp nor Vu TurpsM
TrigppMM VuTur P M p Vu
no M p Un
Ture Me Vu
Mf MerRiemann tenser b
Rivals
Covariant Derivative On nm nm
PnfTLpMH fµV
ftp 1MH TLPPMH qvrPutman MMtT PnMM PnV04Th MH pmVV
PMtua mm Pmf
go.OMT.apmdtgarlmwtg
ofoncg.ws
gas o P gov
0µg oµ4TrapM4 on uT ddp to
PrPu torta in general
for 0 0 0qq.V Ppbvn Rnvs.pt0a0pWuPpPaWrRnvapWTLevi Civita tensor
Identity tensor of gM
Tensor f MinsMtv
V.V
original condo xn new cords In
Iii to IEIIi kEIiosIi.nIT iiJY qx of FINI oxI
ox ON JB 9 2 gyu gyu
JT l n nachoLevi Civita tensor type 9 tenserCmap
FT if Muap mean even oddpermutationof oil 2,3
0 otherwise
totally antisymmetric Enya EmptEarpEµpva
Emma µ 9i9 h exam
EEE i m iE nfmh nE m FE
EMMYMy pgtdet x if Matheny is
an evenherddpermof0 I 2
o otherwise
In.m h m'f m9aag det TgFF detoxify FJ
My µ µIFJ it MMM344 is beyond
Permof 01112,3o otherwise
Levi Civita tenser
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